TS 753 
.T6 

^Opy 1 



DIAMOND DESIGN 



DIAMOND DESIGN 

A STUDY OF THE REFLECTION 

AND REFRACTION OF LIGHT IN 

A DIAMOND 



BY 



MARCEL TOLKOWSKY 

B.Sc, 'A.C.G.I. 



WITH S7 ILLUSTRATIONS 




Xon^on: 

E. & F. N. SPON, Ltd., 57 HAYMARKET, S.W. i 
IWew ll)orft: 

SPON & CHAMBERLAIN, 120 LIBERTY STREET 
I919 



-X?)15 



s 



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CONTENTS 



INTRODUCTION . . . . 


PAGE 

5 


PART L- 


-HISTORICAL 


8 


„ II.- 


-OPTICAL 


. 26 


„ III.- 


-MATHEMATICAL . 


. . 53 




The Rose 


. 59 




The Brilliant 


. . 64 




A. Back . . . 


. . 64 




B. Front . 


. 80 




Faceting 


. 94 



Best Proportions of a Brilliant 97 



DIAMOND DESIGN 

INTRODUCTION 

This book is written principally for students 
of precious stones and jewellers, and more 
particularly for diamond manufacturers 
and diamond cutters and polishers. The 
author will follow the evolution of the 
shape given to a cut diamond, and discuss 
the values of the various shapes and the 
reason for the discarding of the old shapes 
and the practically^ universal adoption of 
the brilliant. • 

It is a remarkable fact that, although the 
art of cutting a diamond has been known for 
more than two thousand years, it is entirely 
empirical, and that, though many keen con- 
temporary minds have been directed upon 
the diamond, and the list of books written 
on that subject increases rapidly, yet 



6 DIAMOND DESIGN 

nowhere] can one find any mathematical 
work determining the best shape for that 
gem. The present volume's chief aim is 
the calculation of that shape. 

The calculations have been made as 
simple as possible, so as not to be beyond 
the range of readers with a knowledge of 
elementary geometry, algebra, and trigo- 
nometry. Where, however, it was found 
that the accuracy of the results would be 
impaired without the introduction of more 
advanced mathematics, these have been 
used, and graphical methods have been 
explained as an alternative. 

The results of the calculations for the form 
of brilliant now in use were verified by 
actual mensuration from well-cut brilliants. 
The measures of these brilliants are given 
at the end of the volume both in a tabulated 
and in a graphical form. It will be seen 
how strikingly near the actual measures are 
to the calculated ones. 

The method used in the present work 
will be found very useful for the design 
of other transparent precious and semi- 



INTRODUCTION 7 

precious stones, although it will be found 
advisable in the case of stones of an agree- 
able colour to cut the gem somewhat 
thicker than the calculations warrant, so 
as to take full advantage of the colour. 
The same remark applies to diamonds of 
some exceptional and beautiful colour, like 
blue or pink, where the beauty or the value 
of the stone increases with the depth of 
its colour. 



Part I 
HISTORICAL 

It is to Indian manuscripts and early Indian 
literature we turn when we want to find 
the origin of diamond cutting, for India 
has always been regarded as the natural 
and ancient home of the diamond. It is 
there that they were first found : up to 
1728, the date of the discovery of the 
Brazilian deposits, practically the whole 
world's supply was derived from Indian 
sources. They are found there in the 
valleys and beds of streams, and also, 
separated from the matrix in which they 
were formed, in strata of detrital matter 
that have since been covered by twelve 
to sixteen feet of earth by the accumula- 
tion of later centuries. Diamonds have 
existed in these deposits within the reach 
of man for many ages, but the knowledge 



HISTORICAL 9 

of the diamond as a gem or as a crystal- 
with exceptional qualities does not go 
back in India to the unfathomable antiquity fo 
which books on diamonds generally refer. 
Ji^f: It was wholly unknown in the Vedic 
period, from which no specific names for 
precious stones are handed down at all.^ 
The earliest systematic reference appears 
to be in the Arthagastra of Kautilya (about 
third century B.C.), where the author 
mentions six kinds of diamonds classified 
according to their mines, and describes 
them as differing in lustre and hardness. 
He also writes that the best diamonds 
should be large, regular, heavy, capable of 
bearing blows, ^ able to scratch metal, 

^ Berthold Laufer, The Diamond : a Study in 
Chinese and Hellenistic Folklore (Chicago, 1915). 

■^ This legend of the indestructibihty of the diamond, 
which reappears in many other places, and to which 
the test of the diamond's capacity of bearing the 
strongest blows was due, has caused the destruction 
of perhaps a very large number of fine stones, -^he 
legend was further embroidered by the remark that 
if the diamond had previously been placed in the 
fresh and still warm blood of a ram, it could then be 
broken, but with great difficulty. This legend was 



10 DIAMOND DESIGN ' 

refractive and brilliant. In the Milinda- 
panha (Questions of King Milinda) (about 
first century B.C.) we read that the diamond 
ought to be pure throughout, and that it 
is mounted together with the most costly 
gems. This is the first manuscript in which 
the diamond is classed as a gem. 

It is therefore permissible to estimate 
with a sufficient degree of accuracy that 
the diamond became known in India during 
the Buddhist period, about the fourth 
century B.C., and that its use as a gem 
dates from that period. ^ 

It is not known with certainty when 
and where the art of grinding or polishing 
diamonds originated. There is as 3^et no 
source of ancient Indian literature in which 
the polishing of diamonds is distinctly set 
forth, although the fact that diamond is 
used for grinding gems generally is men- 
still current in Europe as late as the middle of the 
thirteenth century. The actual fact is that the 
diamond, although exceedingly hard (it is the hardest 
substance known), can easily be split by a light blow 
along a plane of crystallisation. 

^ Laufer, loc. cit. 



HISTORICAL II 

tioned. It is, however, likely that, where ' 
the pohshing of other precious stones was 
accomplished in that manner, that of 
diamonds themselves cannot have been 
entirel}^ unknown. What pohshing there 
was must at first have been limited to 
the smoothing of the faces of the crystals 
as they were found. The first description 
of cut diamonds is given by Tavernier,i 
a French jeweller who travelled through 
India, and to whom we owe most of our 
knowledge of diamond cutting in India in 
the seventeenth century. At the time of 
his visit (1665) the Indians were pohshing 
over the natural faces of the crystal, and 
preferred, therefore, regularly crystaUised 
gems. They also used the knowledge they 
had of grinding diamonds to remove faulty 
places hke spots, grains, or glesses. If the 
fault was too deep, they attempted to hide 
it by covering the surface under which it 
lay with a great number of small facets. 
It appears from Tavernier's writings that 

1 Tavernier, Voyage en Turqiiie, en Perse et mix 
Indes (1679). 



12 DIAMOND DESIGN 

there were also European polishers in 
India at that time, and that it was to them 
the larger stones were given for cutting. 
Whether they had learnt the art inde- 
pendently or from Indians and attained 
greater proficiency than they, or whether 
they were acting as instructors and teaching 
the Indians a new or a forgotten art, is 
uncertain. Both views are equally likely 
in the present state of research upon that 
subject : at the time of Tavernier's visit, 
diamond cutting had been known in Europe 
for more than two centuries. 

Among the several remarkable gems that 

Tavernier describes, 
the most noteworthy 
is the one known as 
the Great Mogul. 
This diamond was of 
a weight of 280 cts. 
and was cut as 
sketched in f^g. i. 
The polishing was 
the work of a Venetian, Hortensio Borgis, 
to whom it was given for that purpose by 




Fig. 



HISTORICAL 



13 




its owner, the Great Mogul Aurung Zeb, of 

Delhi. This kind of cut is characteristic 

of most of the 

large Tn d i a n 

stones, such as 

theOrlow(fig.2), 

which is now the 

largest diamond 

of the Russian 

crown jewels and weighs 193! cts. The Koh-i- 

Noor (fig. 3), now among the British crown 

jewels, was of a somewhat similar shape 

before recutting. 
It weighed then 
186 cts. 

Ta vernier also 



Fig. 2. 




Fig. 3. 



mentions several 
other types of 
cut which he 
met in India. 
The Great Table (fig. 4), which he saw in 
1642, weighed 242 cts. Both the Great Table 
and the Great Mogul seem to have dis- 
appeared : it is not known what has become 
of them since the seventeenth century. 



14 



DIAMOND DESIGN 




Fig. 4. 



Various other shapes are described, such 
as point stones, thick stones, table stones 

(fig. 5), etc. But the 
chief characteristic 
remains : all these 
diamonds have been 
cut with one aim con- 
stantly in view — ^how 
to pohsh the stone with the smallest 
possible loss of weight. As a consequence 
the polishing was generally accomplished 
by covering the surface of the 
stone with a large number of 
facets, and the original shape 
of the rough gem was, as far as 
possible, left unaltered. 

It was mentioned before that 
the art of diamond polishing 
had already been known in 
Europe for several centuries when Tavernier 
left for India. We have as yet no cer- 
tain source of information about diamond 
cutting in Europe before the fourteenth 
century. The first reference thereto men- 
tions that diamond polishers were work- 



\ / 


/ 








\ 




Fig. 



HISTORICAL 15 

ing ill Nurnberg (Germany) in 1375, where 
they formed a guild of free artisans, to 
which admission was only granted after 
an apprenticeship of five to six years. ^ 
We do not know, however, in what 
shape and by what method the stones 
were cut. ^ 

It is in the fifteenth century that Euro- 
pean diamond cutting begins to become 
more definite, more characteristic. And it 
is from that time that both on its technical 
and artistic sides progress is made at a 
rate, slow at first, but increasing rapidly 
later. 

It is not difficult to find the chief reason 
for that change. 

Up to that time, diamonds had almost 
exclusively been used by princes or priests. 
To princes they were an emblem of power 
and wealth — ^in those days diamonds were 
credited with extraordinary powers : they 
were supposed to protect the wearer and 
to bring him luck. Princes also found 
them convenient, as they have great value 
^ Jacobson's Technologisches Worterbuch (1781). 



i6 DIAMOND DESIGN 

for a very small weight, and could easily 
be carried in case of flight. Priests used 
them in the ornaments of temples or 
churches ; they have not infrequently been 
set as eyes in the heads of statues of 
Buddha. 

In the fifteenth century it became the 
fashion for women to wear diamonds as 
jewels. This fashion was started by Agnes 
Sorel (about 1450) at the Court of Charles 
VII of France, and gradually spread from 
there to all the Courts of Europe. 

This resulted in a very greatly increased 
demand, and gave a strong impulse to the 
development of diamond polishing. The 
production increased, more men applied 
their brains to the problems that arose, 
and, as they solved them and the result 
of their work grew better, the increas- 
ing attractiveness of the gem increased 
the demand and gave a new impulse to 
the art. 

At the beginning of the fifteenth century 
a clever diamond cutter named Hermann 
established a factory in Paris, where his 



HISTORICAL ' 17 

work met with success, and where the 
industry started developing. 

In or about 1476 Lodewyk (Louis) van 
Berquem, a Flemish polisher of Bruges, 
introduced absolute symmetry in the dis- 
position of the facets, and probably also 
improved the process of polishing. Early 
authors gave credence to the statement 
of one of his descendants, Robert van 
Berquem, 1 who claims that his ancestor 
had invented the process of polishing the 
diamond by its own powder. He adds : 
'' After having ground off redundant 
material from a stone by rubbing it against 
another one (the process known in modern 
practice as ' bruting ' or cutting), he col- 
lected the powder produced, by means of 
which he polished the diamond on a mill 
and certain iron wheels of his invention.'' 

^ Robert de Berquem, Les merveilles des Indes : 
Traite des pierres precieuses (Paris, in-40, 1669), 
p. 12 : " Louis de Berquem Tun de mes ayeuls a 
trouve le premier Tinvention en mil quatre cent 
soixante-seize de les tailler avec la poudre de diamant 
meme. Auparavant on fut contraint de les mettre 
en oeuvre tels qu'on les rencontrait aux Indes, c'est- 

2 



i8 DIAMOND DESIGN 

As has already been shown, we know now 
that diamonds were polished at least a 
century before Lodewyk van Berquem lived. 
And as diamond is the hardest substance 
known, it can only be polished by its own 
powder. Van Berquem cannot thus have 
invented that part of the process. He may 
perhaps have introduced some important 
improvement like the use of cast-iron 
polishing wheels, or possibly have discovered 
a more porous kind of cast iron — one on 
which the diamond powder finds a better 
hold, and on which polishing is therefore 
correspondingly speedier. 

a-dire tout a fait bruts, sans ordre et sans^ grace, 
sinon quelques faces au hasard, irregulieres et mal 
polies, tels enfin que la nature les produit. II mit 
deux diamants sur le ciment et apres les avoir egrises 
Tun contre I'autre, il vit manifestement que par Ic 
moyen de la poudre qui en tombait et a Faide du 
moulin et certaines roues de fer qu'il avait inventees, 
il pourrait venir a bout de les polir parfaitement, 
meme de les tailler en telle maniere qu'il voudrait. 
Charles devenu due de Bourgogne lui mit trois grands 
diamants pour les tailler avantageusement selon 
son adresse. II les tailla aussitot, I'un epais, Tautre 
faible et le troisieme en triangle et il y reussit si bien 
que le due, ravi d'une invention si surprenante, lui 
donna 3000 ducats de recompense." 



HISTORICAL 



19 



What Van Berquem probably did ori- 
ginate is, as already stated, rigid symmetry 
in the design of the cut stone. The intro- 
duction of the shape known as pendeloque 
or briolette is generally ascribed to him. 
The Sancy and thfe Florentine, which are 
both cut in this shape, have been said 




Fig. 6. 

by some to have been polished by him. 
The Sancy (53 J cts.) belongs now to the 
Maharaja of Guttiola, and the Florentine 
(fig. 6), which is much larger (133 1 cts.), 
is at present among the Austrian crown 
jewels. The history of both these gems 
is, however, very involved, and they may 
have been confused at some period or 
other with similar stones. That is why 
it is not at all certain that the}^ were 



20 DIAMOND DESIGN 

the work of Van Berquem. At any rate, 
they are typical of the kind of cut he 
introduced. 

The pendeloque shape did not meet with 
any very wide success. It was adopted 
in the case of a few large stones, but was 
gradually abandoned, and is not used to 
any large extent nowadays, and then in 
a modified form, and only when the shape 
of the rough stone is especialh^ suitable. 
This unpopularity was largely due to the 
fact that, although the loss of weight in 
cutting was fairly high, the play of light 
within the stone did not produce sufficient 
fire or brilliancy. 

About the middle of the sixteenth century 
a new form of cut diamond was introduced. 
It is known as the rose or rosette, and was 
made in various designs and proportions 
(figs. 7 and 8). The rose spread rapidly 
and was in high vogue for about a century, 
as it gave a more pleasant effect than the 
pendeloque, and could be cut with a much 
smaller loss of weight. It was also found 
very advantageous in the polishing of flat 



HISTORICAL 21 

pieces of rough or split diamond. Such 
material is even ' now frequently cut into 
roses, chiefly in the smaller sizes. 




Fig. 7. 

In the chapter upon the design of 
diamonds it will be shown that roses have 
to be made thick (somewhat thicker than 
in fig. 7) for the loss of light to be small, 
and that the flatter the 
rose the bigger the loss 
of light. It will also be 
seen there that the fire 
of a rose cannot be 
very high. These faults 
caused the rose to be 
superseded by the 
brilliant. f^°- ^• 

We owe the introduction of the brilliant 
in the middle of the seventeenth centurv 
to Cardinal Mazarin — or at any rate to 
his influence. As a matter of fact, the first 
brilliants were known as Mazarins, and were 




22 



DIAMOND DESIGN 



of the design of fig. 9. They had sixteen 
facets, excluding the table, on the upper side. 




Fig. 9, 



They are called double-cut brilliants. Vincent 
Peruzzi, a Venetian polisher, increased the 
number of facets from sixteen to thirty- 




FiG. 10. 



two (fig. 10) (triple-cut brilliants), thereby 
increasing very much the fire and brilliancy 
of the cut gem, which were already in the 
double-cut brilliant incomparably better 



HISTORICAL 23 

than in the rose. Yet diamonds of that 
cut, when seen nowadays, seem exceedingly 
dull compared to modern-cut ones. This 
dullness is due to their too great thickness, 
and to a great extent also to the difference 
in angle between the corner facets and the 
side facets, so that even if the first were 




Fig. II 

polished to the correct angle (which they 
were not) the second would be cut too 
steeply and give an effect of thickness. 
Old-cut brilliants, as the triple-cut brilliants 
are generally called, were at first modified 
by making the size and angle of the facets 
more uniform (fig. 11), this bringing about 
a somewhat rounder stone. With the in- 
troduction of mechanical bruting or cutting 
(an operation distinct from polishing ; see 



24 DIAMOND DESIGN 

p. 17) diamonds were made absolutely 
circular in plan (fig. 37). The gradual 
shrinking-in of the corners of an old-cut 
brilliant necessitated a less thickly cut 
stone with a consequent increasing fire and 
life, until a point of maximum brilliancy 
was reached. This is the present-day 
brilliant.^ 

Other designs for the brilliant have been 
tried, mostly attempts to decrease the loss 
of weight in cutting without impairing the 
brilliancy of the diamond, but they have 
not met with success. 

We may note here that the general 
trend of European diamond polishing as 
opposed to Indian is the constant search 
for greater brilliancy, more life, a more 

^ Some American writers claim that this change 
from the thick cut to that of maximum brilliancy was 
made by an American cutter, Henry D. Morse. It 
was, however, as explained, necessitated by the absolute 
roundness of the new cut. Mr Morse may have 
invented it independently in America. But it is 
highly probable that it originated where practically 
all the world's diamonds were polished, in Amsterdam 
or Antwerp, where also mechanical bruting was first 
introduced. 



HISTORICAL 



25 



vivid fire in the diamond, regardless of 
the loss of weight. The weight of diamond 
removed by bruting and by polishing 
amounts even in the most favourable cases 
to 52 per cent, of the original rough weight 
for a perfectly cut brilliant. In the next 
chapters the best proportions for a brilliant 
will be calculated without reference to the 
shape of a rough diamond, and it will be 
seen how startlingly near the calculated 
values the modern well-cut brilliant is 
polished. 



Part II 

OPTICAL 

It is to light, the play of Ught, its reflection 
and its refraction, that a gem owes its 
brilliancy, its fire, its colour. We have 
therefore to study these optical properties 
in order to be able to apply them to the 
problem we have now before us : the cal- 
culation of the shape and proportions of 
a perfectly cut diamond. 

Of the total amount of light that falls 
upon a material, part is returned or re- 
flected ; the remainder penetrates into it, 
and crosses it or is absorbed by it. The 
first part of the Ught produces what is 
termed the ''lustre" of the material. 
The second part is completely absorbed 
if the material is black. If it is partly 
absorbed the material will appear coloured, 

26 



OPTICAL 27 

and if transmitted unaltered it will appear 
colourless. 

The diamonds used as gems are generally 
colourless or only faintly coloured ; it can 
be taken that all the light that passes into 
the stones passes out again. The lustre 
of the diamond is peculiar to that gem, 
and is called adamantine for that reason. 
It is not found in any other gem, although 
zircon and demantoid or olivine have a 
lustre approaching somewhat to the ada- 
mantine. 

In gem stones of the same kind and of 
the same grade of polish, we may take it 
that the lustre only varies with the area 
of the gem stone exposed to the light, and 
that it is independent of the type of cut 
or of the proportions given to the gem (in 
so far as they do not affect the area) ; this 
is why gems where the amount of light 
that is reflected upon striking the surface 
is great, or where much of the light that 
penetrates into the stone is absorbed and 
does not pass out again, are frequently cut 
in such shapes as the cabochon (fig. 12), 



28 DIAMOND DESIGN 

so as to get as large an area as possible, 
and in that way take full advantage of 
the lustre. 

In a diamond, the amount of light re- 
flected from the surface is much smaller 
that that penetrating into the stone ; more- 
over, a diamond is practically perfectly 
transparent, so that all the light that passes 
into the stone has to pass out again. This 




Fig. 12. 

is why lustre may be ignored in the working 
out of the' correct shape for a diamond, 
and why any variation in the amount of 
light reflected from the exposed surface 
due to a change in that surface may be 
considered as negligible in the calculations. 
The brilliancy or, as it is sometimes 
termed, the ''fire'' or the ''life'' of a gem 
thus depends entirely upon the play of 
light in the gem, upon the path of rays 
of light in the gem. If a gem is so cut or 
designed that every ray. of light passing 



OPTICAL 29 

into it follows the best path possible for 
producing pleasing effects upon the eye, 
then the gem is perfectly cut. The whole 
art of the lapidary consists in proportion- 
ing his stone and disposing his facets so 
as to ensure this result. 

If we want to design a gem or to calculate 
its best shape and proportions, it is clear 
that we must have sufficient knowledge to 
be able to work out the path of any ray of 
light passing through it. This knowledge 
comprises the essential part of optics, and 
the laws which have to be made use of 
are the three fundamental ones of reflection, 
refraction, and dispersion. 

Reflection 

Reflection occurs at the surface which 
separates two different substances or media ; 
a portion or the whole of the light striking 
that surface is thrown back, and does not 
cross over from one medium into another. 
This is the reflected light. There are dif- 
ferent kinds of reflected light according to 
the nature of the surface of reflection. If 



30 DIAMOND DESIGN 

that surface is highly polished, as in the 
case of mirrors, or polished metals or gems, 
the reflection is perfect and an image is 
formed. The surface may also be dull 
or matt to a greater or smaller extent (as 
in the case of, say, cloth, paper, or pearls). 
The reflected light is then more or less 
scattered and diffused. 

It is the first kind of reflection that is of 
importance to us here, as diamond, owing 
to its extreme hardness, takes a very high 
grade of polish and keeps it practically^ 
for ever. 

The laws of reflection can be studied very 
simply with a few pins and a mirror placed 
at right angles upon a flat sheet of paper. 

A plan of the arrangement is shown in 
fig. 13. The experiment is as follows : — 

I. A straight line A B is drawn upon 
the paper, and the mirror is stood on the 
paper so that the plane of total reflection 
(i.e. the silvered surface) is vertically over 
that line. Two pins P and Q are stuck 
anyhow on the paper, one as near the 
mirror and the other as far away as possible. 



OPTICAL 



^.i 



Then the eye is placed in line with FO at 
I, so that Q is hidden by P. Without 
moving the eye, two more pins R and S 
are inserted, one near to and the other far 
from the mirror, in such positions that 




Fig. 13. 

their images appear in the mirror to lie 
along PQ continued. 

If the eye is now sighted from position 
2 along S R, Q and P will appear in the 
mirror to lie on S R continued. 

The mirror is now removed, P and S R 
are joined and will be found to intersect 
on A B at M. If a perpendicular M N be 



32 DIAMOND DESIGN 

erected on AB at that point, the angles 
N M P and N M S will be found equal. 

The above experiment may be repeated 
along other directions, but keeping the pin 
S at the same point. The line of sight 
will now lie on P' Q', and the angles 
between P' Q', S R' and the normal will 
again be found equal. 

In the first experiment S appeared to 
lie on the continuation of P Q, in the second 
it appears to be situated on P' Q' produced. 
Its image is thus at the intersection of these 
two lines, at L. It can easily be proved 
by elementary geometry (from the equality 
of angles) that the image L of the pin is 
at the same distance from the mirror as 
the pin S itself, and is of the same size. 

II. If the pins P, Q, R, S in the first 
experiment be placed so that their heads 
are all at the same height above the plane 
sheet of paper, and the eye be placed in a 
line of sight with the heads P, Q, the images 
of the heads R, S in the mirror will be 
hidden b}/ the head of pin P. 

The angle N M P (position I) is called 



OPTICAL 33 

angle of incidence, and the angle N M S 
angle of reflection. 

The laws of reflection (verified by the 
above tests) can now be formulated as 
follows : — 

I. The angle of reflection is equal to the 
angle of incidence., 

IL The paths of the incident and of the 
reflected ray lie in the same plane. 

From I it follows, as shown, that 

in. The image formed in a plane re- 
flecting surface is at the same distance 
from that surface as the object reflected, 
and is of the same size as the object. 

. JRefraction 

When light passes from one substance 
into another it suffers changes which are 
somewhat more complicated than in the 
case of reflection. Thus if we place a coin 
at the bottom of a tumbler which we fill 
with water, the coin appears to be higher 
than when the tumbler was empty; also, 
if we plunge a pencil into the water, it 
will seem to be bent or broken at the surface, 

3 



34 



DIAMOND DESIGN 



except in the particular case when the 
pencil is perfectly vertical. 

We can study the laws of refraction in 
a manner somewhat similar to that adopted 
for the reflection tests. Upon a fiat sheet 
of paper (fig. 14) we place a fairly thick 




Fig. 14. 

rectangular glass plate with one of its 
edges (which should be poHshed perpendi- 
cularly to the plane of the paper) along a 
previously drawn line A B. We place a 
pin, P, close to the edge AB of the glass 
plate and another, Q, close to the further 
edge. Looking through the surface A B, 



OPTICAL 35 

we place our eye in such a position that 
the pin Q as viewed through the glass is 
covered by pin P. Near to the eye and 
on the same hne of sight we stick a third 
pin R, which therefore covers pin P. The 
glass plate is now removed. PQ and PR 
are joined, a perpendicular to AB, MM', 
is erected at P, and a circle of any radius 
drawn with P as centre. This circle cuts 
PQ at K and RP at L. LM and KM' 
are drawn perpendicular to MM', L M and 

K M' are measured and the ratio tfitt? 

found. 

The experiment is repeated for different 

positions of P and Q and the corresponding 

.. LM , , 
ratio j^^, calculated. It will be found that 

for a given substance (as in this case glass) 
this ratio is constant. It is called the 
index j4 refraction, and generally repre- 
sented by the letter n. 

Referring to fig. 14, we note that as 

P K = P L = radius of the circle. 



36 DIAMOND DESIGN 

we can write 

LM 

sin R P M 





LM 


LM 


PL 


KM' 


~KM' 



sin Q P M' 
PK 

Writing the angle of incidence R P M as i, 
and the angle of refraction Q P M' as f , this 

equation becomes 

sin 2^ , X 

n=-. — . . . I 
sm r 

or 

n sin r = sin i . . (2) 

In this case the incident ray is in air, 
the index of refraction of which is very 
nearly unity. With another substance it 
can be shown that equation (2) becomes 

n sin f = n' sin i , . (3) 

where n' is the index of refraction of that 
substance. 

It can be seen easily, and in a way similar 
to that used with reflection {i.e. sighting 
along the heads of the pins), that, in re- 
fraction also : 

The paths of the incident and of the 
refracted ray lie in the same plane. 



OPTICAL 



37 



Of two substances with different index' 
qf refraction, that which has the greater 
index of refraction is called optically denser. 
In the experiment the light passed from air 
to glass, which is of greater optical density. 
Let us now consider the reverse case, i.e. 

N' 

A' 




when light passes from one medium to 
another less dense optically. Suppose a 
beam of light AO (fig. 15) with a small 
angle of incidence passes from water into 
air. At the surface of separation a small 
proportion of it is reflected to A'' (as we 
have seen under reflection) . The remainder 
is refracted in a direction O A' which is 



38 DIAMOND DESIGN 

more divergent from the normal N O N' 
than AO. 

Suppose now that the angle AON gradu- 
ally increases. The proportion of reflected 
light also increases, and the angle of re- 
fraction N'OA' increases steadily and at 
a more rapid rate than N O A, until for a 
certain value of the angle of incidence 
BON the refracted angle will graze the 
surface of separation. It is clear that under 
these conditions the amount of light which 
is refracted and passes into the air is 
zero. If the angle of incidence is still 
greater, as at CON, there is no re- 
fracted ray, and the whole of the light is 
reflected into the optically denser medium, 
or, as it is termed, total reflection then 
occurs. The angle B O N is called critical 
angle, and can easily be calculated by (3) 
when the refractive indices n and n' are 
known. It will be noted that when the 
angle of incidence attains its critical 
value i' , the angle of refraction becomes 
a right angle, i.e. its sine becomes equal 
to unity. 



OPTICAL 39 

Substituting in (3) 

n sin r = n^ sin ^' 
sin r — I 

sm ^ =- -7 . • • (4) 
n 

Or, if the less dense medium be air, 

n = 1 

sini' =^^ . . (5) 

n 

This formula (5) is very important in 
the design of gems, for by its means the 
critical angle can be accuratety calculated. 
A precious stone, especially a colourless 
and transparent one like the diamond, is 
cut to the best advantage and with the 
best possible effect when it sends to the 
spectator as strong and as dazzling a beam 
of light as possible. Now a gem, not being 
in itself a source of light, cannot shine with 
other than reflected light. The maximum 
amount of light will be given off by the gem 

* No mention is made here of double refraction, 
as the diamond is a singly refractive substance, and 
it was considered unnecessary to introduce irrelevant 
matter. 



40 



DIAMOND DESIGN 



if the whole of the light that strikes it is 
reflected by the back of the gem, i.e. by 
that part hidden by the setting, and sent 
out into the air by its front part. The 
facets of the stone must therefore be so 
disposed that no light that enters it is let 
out through its back, but that it is wholly 
reflected. This result is obtained by having 
the facets inclined in such a way that all 
the light that strikes them does so at an 
angle of incidence greater than the critical 
angle. This point will be further dealt with 
in a later chapter. 

The following are a few indices of refrac- 
tion which may be useful or of interest : — 



Water . 


1-33 


Crown glass 


. .1-5 approx 


Quartz 


. I-54-I-55 


Flint glass 


. 1-576 


Colourless strass 


1-58 


Spinel 


172 


Almandine 


179 


Lead borate 


1.83 


Demantoid 


. 1-88 



OPTICAL 41 

Lead silicate . . 2-12 
Diamond . . . 2*417 . (6) 

These indices have, of course, been found 
by methods more accurate than the tests 
described. One of these methods, one 
particularly suitable for the accurate de- 
termination of the indices of refraction in 
gems, will be explained later. 

With this value for the index of re- 
fraction of diamond, the ^ critical angle 
works out at 



sm^ 



I 
n 

I 



= -4136 



2-417 
i = sin~^ '4136 
i =- 24° 26' . . . (7) 

This angle will be found very important. 

Dispersion 

What we call white light is made up of 
a variety of different colours which produce 
white by their superposition. It is to the 



42 DIAMOND DESIGN 

decomposition of white light into its com- 
ponents that are due a variety of beautiful 
phenomena like the rainbow or the colours 
of the soap bubble — and, it may be added, 
the '' fire " of a diamond. 

The index of refraction is found to be 
different for light of different colours, red 
being generally refracted least and violet 
most, the order for the index of the various 
colours being as follows : — 

Red, orange, yellow, green, blue, indigo, 
violet. 

Note. — In the list given above the 
index of refraction is that of the 
yellow light obtained by the incan- 
descence of a sodium salt. This colour 
is used as a standard, as it is very 
bright, very definite, and easily pro- 
duced. 
If white light strikes a glass plate with 
parallel surfaces (fig. 20) the different colours 
are refracted as shown when passing into 
the glass. Now for every colour the angle 
of refraction is given by (equation (2)) 
# sin r = sin i. 



OPTICAL 



43 



When passing out of the glass, the angle 
of refraction is given by 



n sm I = sm r 



As the faces of the glass are parallel, i' — r. 

Therefore, / = i, and the ray when leaving 
the glass is parallel to its original direction. 
The various colours will thus follow parallel 




Fig. I 6. 



paths as shown in fig. i6, and as they are 
very near together (the dispersion is very 
much exaggerated), they will strike the eye 
together and appear white. This is why 
in the pin experiments on refraction, dis- 
persion was not apparent to any extent. 

If, instead of using parallel surfaces as in 
a glass plate, we place them at an angle, 
as in a prism, light falling upon a face of 



44 DIAMOND DESIGN 

the prism will be dispersed as shown in 
fig. 17 ; and, when leaving by another 
face, the light, instead of combining to form 
white (as in a plate) , is still further dispersed 
and forms a ribbon of lights of the different 




Fig. 17. 

colours, from red to violet. Such a ribbon 
is called a spectrum. The colours of a 
spectrum cannot be further decomposed by 
the introduction of another prism. 

The difference between the index of 
refraction of extreme violet light and that 
of extreme red is called dispersion.^ Dis- 

1 Generally two definite points on the spectrum 
are chosen ; the values given here for gems are those 
between the B and G lines of the solar spectrum. 



OPTICAL 



45 



persion, on the whole, increases with the 
refractive index, although with exception. 
The dispersion of a number of gems and 
glasses is given below : — 



Quartz 


• -013 


Sapphire 


. -018 


Crown glass 


. '019 


Spinel 


. -020 


Almandine . 


. -024 


Flint glass . 


. -036 


Diamond 


. -044 


Demantoid . 


. -057 



The greater the dispersion of a medium, 
other things being equal, the greater the 
difference between the angles of refraction 
of the various colours, and the further 
separated do they become. It is to its 
very high dispersion (the greatest of all 
colourless gem-stones) that the diamond 
owes its extraordinary '' fire." For when 
a ray of light passes through a well-cut 
diamond, it is refracted through a large 
angle, and consequently the colours of the 
spectrum, becoming widely separated, strike 



46 DIAMOND DESIGN 

a spectator's eye separately, so that at 
one moment he sees a ray of vivid blue, 
at another one of flaming scarlet or one of 
shining green, while perhaps at the next 
instant a beam of purest white may be 
reflected in his direction. And all these 
colours change incessantly with the slightest 
motion of the diamond. 

The effect of refraction in a diamond can 
be shown very interestingly as follows : — 
A piece of white cardboard or fairly stiff 
paper with a hole about half an inch in 
diameter in its centre is placed in the 
direct rays of the sun or another source 
of light. The stone is held behind the 
paper and facing it in the ray of light which 
passes through the hole. A great number 
of spots of the most diverse colours appear 
then upon the paper, and with the slightest 
motion of the stone some vanish, others 
appear, and all change their position and 
their colour. If the stone is held with the 
hand, its slight unsteadiness will give a 
startling appearance of life to the image 
upon the paper. This life is one of the 



OPTICAL 47 

chief reasons of the diamond's attraction, 
and one of the main factors of its beauty. 

Measurement of Refraction 

In the study of refraction it was pointed 
out that the manner by which the index 
of refraction was calculated there, although 
the simplest, was both not sufficiently 
accurate and unsuitable for gem-stones. 
One of the best methods, and perhaps the 
one giving the most correct results, is that 
known as method of minimum deviation. 
Owing to the higher index of refraction of 
diamond it is especially suitable in its case, 
where others might not be convenient. 
The theory of that method is as follows : — 
Let A B C be the section of a prism of the 
substance the refraction index of which 
we want to calculate (fig. i8). A source 
of light of the desired colour is placed at 
R, and sends a beam R I upon the face A B 
of the prism. The beam RI is broken, 
crosses the prism in the direction IF, is 
again broken, and leaves it along F R'. 
Supposing now that we rotate prism ABC 



48 



DIAMOND DESIGN 



about its edge A. The direction of F R'' 
changes at the same time ; we note that as 
we gradually turn the prism, V R' turns 
in a certain direction. But if we go on 
turning the prism, F R' will at a certain 
moment stop and then begin to turn in 




Fig. iS. 



the reverse direction, although the rotation 
of the prism was not reversed. We also 
note that at the moment when the ray is 
stationary the deviation has attained its 
smallest value. It is not difficult to prove 
that this is the case when the ray of light 
passes through the prism symmetrically, 
i.e. when angles i and i' (fig. i8) are equal. 



OPTICAL 49 

Let A M be a line bisecting the angle A. 
Then I T is perpendicular to A M. Let R I 
be produced to Q and R' I' to O. They meet 
on A M and the angle Q O R' is the deviation d 
[i.e. the angle between the original and 
the final direction of the light passing 
through the prism). 

Therefore OIF-: \d. 

Draw the normal at I, N N'. 

Then 

MIN' = IAM=-i^ 

if a be the angle B A C of the prism. 
Now by equation (i) 

sin i 

n = - — 

sm r 

i::::.NIR-:OIN'-OIM+MIN' 

^ Id+la =^{d+a) 

r - M I N' = 1^ 



therefore 



sin Md-\-a) ,n\ 

n = ^A_^ — L . . (8) 

sm fa 



The index of refraction can thus be cal- 
culated if the angles d and a are known. 
These are found by means of a spectroscope. 



50 DIAMOND DESIGN 

This instrument consists of three parts : the 
collimator, the table, and the telescope. The 
light enters by the collimator (a long brass 
tube fitted with a slit and a lens) passes 
through the prism which is placed on the 
table, and leaves by the telescope. The 
colUmator is usually mounted rigidly upon 
the stand of the instrument. Its function 
is to determine the direction of entry of 
the light and to ensure its being parallel. 
Both the table and the telescope are 
movable about the centre of the table, and 
are fitted with circular scales which are 
graduated in degrees and parts of a de- 
gree, and by means of which the angles 
are found. 

Now two facets of a stone are selected, 
and the stone is placed upon the table so 
that these facets are perpendicular to the 
table. The angle a of the prism, i.e. the 
angle between these facets, can be found 
by direct measurement with a goniometer 
or also by the spectroscope. The angle d 
is found as follows : — The position of the 
stone is arranged so that the light after 



OPTICAL 



51 



passing through the coUimator enters it 
from one selected facet and leaves it by the 
other. The telescope is moved mitil the 
spectral image of the source of light is 
found. The table and the stone are now 
rotated in the direction of minimum devia- 




F1G/19. 

tion, and at the same time the telescope 
is moved so that the image is kept in view. 
We know that at the point of minimum 
deviation the direction of motion of the 
telescope changes. When this exact point 
is reached the movements of the stone 
and of the telescope are stopped, and the 



52 DIAMOND DESIGN 

reading of the angle of deviation d is taken 
on the graduated scale. 

The values of a and d are now introduced 

in equation (8) : 

^^ _ sin|(a+^) 
sin \a 

and the value of n calculated with the help 
of sine tables or logarithms. 
The values for diamond are 

;^ = 2'4i7 for sodium light 

Dispersion = ^red— ^vioiet == •044- 



Part III 
MATHEMATICAL 

In the survey of the history of diamond 
cutting, perhaps the most remarkable fact 
is that so old an art should have progressed 
entirely by trial and error, by gradual 
correction and slow progress, by the almost 
accidental elimination of faults and intro- 
duction of ameliorations. We have traced 
the history of the art as far back as 1375, 
when the earliest recorded diamond manu- 
factory existed, and when the polishers 
had already attained a high degree of guild 
organisation. We have every reason to 
believe that the process of diamond polish- 
ing was known centuries before. And yet 
all these centuries, when numerous keen 
minds were directed upon the fashioning 
of the gem, have left no single record of 

53 



54 DIAMOND DESIGN 

any purposeful planning of the design of 
the diamond based upon fundamental optics. 
Even the most bulky and thorough con- 
temporary works upon the diamond or 
upon gems generally rest content with 
explaining the basic optical principles, and 
do no more than roughly indicate how these 
principles and the exceptional optical pro- 
perties of the gem explain its extraordinary 
brilliancy ; nowhere has the author seen 
calculations determining its best shape and 
proportions. It is the purpose of the 
present chapter to establish this shape and 
these proportions. The diamond will be 
treated essentially as if it were a worth- 
less crystal in which the desired results 
are to be obtained, i.e. without regard 
to the great value which the relation 
between a great demand and a very 
small supply gives to the least weight of 
the material. 

It is useful to recall here the principles 
and the properties which will be used in 
the calculations. 



MATHEMATICAL 55 

Reflection 

1. The angles of incidence and of re- 
flection are equal. 

2. The paths of the incident and of the 
reflected ray lie in the same plane. 

Refraction 

1. When a ray of light passes from one 
medium into a second of different density, 
it is refracted as by the following equation : 

:^ sin f = n' sin i . . (3) 

where r = angle of refraction. 
i — angle of incidence. 
n = index of refraction of the second 

medium. 
n' = index of refraction of the first 
medium. 

If the first medium is air, n' = 1, and 
equation becomes 

nsinr = sin i . . (2) 

2. When a ray of light passes from one 
medium into another optically less dense, 
total reflection occurs for all values of the 
angle of incidence above a certain critical 



56 DIAMOND DESIGN 

value. This critical angle is given by 
equation 

' smt = - . . (4) 

n 

Or, if the less dense medium be air, 

sin.-' = ^, .• . (5) 

3. The paths of the incident and of the 
refracted ra}^ lie in the same plane. 

Dispersion 

When a ray of light is refracted, dis- 
persion occurs, i.e. the ray is split up into 
a band or spectrum of various colours, 
owing to the fact that each colour has a 
different index of refraction. The disper- 
sion is the difference between these indices 
for extreme rays on the spectrum. 



Data 




In a diamond : 




Index of refraction : n = 


= 2-417 


sodium light) 




dispersion : S = 


= -044 


critical angle : i' = 


= 24° 26' 



(for a 



(7) 



MATHEMATICAL 57 

DETERMINATION OF THE BEST ANGLES 
AND THE BEST PROPORTIONS 

, Postulate. — ^The design of a diamond or 
of any gem-stone must be symmetrical 
about an axis, for symmetry and regularity 
in the disposition of the facets are essential 
for a pleasing result. 

Let us now consider a block of diamond 
bounded by polished surfaces, and let us 
consider the effect on the path of light of 
a gradual change in shape ; we will also 
observe the postulate and keep the block 
symmetrical about its axis. 

Let us take as first section one having 
parallel faces (fig. 20), and let MM' be its 
axis of symmetry. Let us for convenience 
place the axis of symmetry vertically in 
all future work, so that surfaces crossing it 
are horizontal. 

Consider a ray of light S P striking face 
A B. It will be refracted along P Q and 
leave by Q R, parallel to S P (as we have 
seen in studying dispersion) . We also know 
that if NN' is the normal at Q, angles 



58 



DIAMOND DESIGN 



N Q P and Q P M' are equal. Therefore, for 
total reflection, 

O P M' == 24° 26', 

but at that angle of refraction the angle of 
incidence S P M becomes a right angle and 
no light penetrates into the stone. It is 
thus obvious that parallel faces in a gem 




are very unsatisfactory, as all the light 
passing in by the front of a gem passes out 
again by the back without any reflection. 

We can avoid parallelism by inclining 
either the top or the bottom faces at an 
angle with the direction AB. In the first 
case we obtain the shape of a rose-cut 
diamond and in the second case that of a 



MATHEMATICAL 



59 



brilliant cut. We will examine the rose' 
cut in the first instance. 



The Rose 

Consider (fig. 21) a section having the 
bottom surface horizontal, and let us incline 




the top surface A B at an angle a with it. 
To maintain symmetry, another surface 
BC is introduced. We have now to find 
the value of a for which total reflection 
occurs at A C. Now for this to be the case, 
the minimum angle of incidence upon AC 
must be 24° 26'. Let us draw such an 
incident ray P Q. To ensure that no light 



6o DIAMOND DESIGN 

is incident at a smaller angle, we must make 
the angle of refraction at entry 24° 26' and 
arrange the surface of entry as shown, A B, 
for we know that then no light will enter 
at an angle more oblique to A B or more 
vertical to A C. This gives to a a value of 
twice the critical angle, i.e. 48° 52'. Such 
a section is very satisfactory indeed as 
regards reflection, as, owing to its deriva- 
tion, all the light entering it leaves by the 
front part. Is it also satisfactory as regards 
refraction ? 

Let us follow the path of a ray of light of 
any single colour of the spectrum, S P Q R T 
(fig. 22). Let i and r be the angles of 
incidence upon and of refraction out of the 
diamond. 

At Q, P Q N -- R Q N, and therefore in 
triangles A P Q and R C Q 

angle A Q P == angle COR. 

Also by symmetry A == C, 
therefore 

angle A P Q - angle C R Q ; 
it follows that i = r. 



MATHEMATICAL 



6i 



As the angle i is the same for all colours 
of a white ray of light, the various colours 
will emerge parallel out of the diamond 
and give white light. This is the funda- 
mental reason of the unpopularity of the 
rose ; there is no fire. 




This effect may be remedied to a small 
extent by breaking the inclined facet (figs. 
23 and 24), so that the angle be not the same 
at entry as at exit. This breaking is harm- 
ful to the amount of hght reflected which- 
ever way we arrange it ; if we steepen the 
facet near the edge, there is a large propor- 
tion of Hght projected backwards and being 



62 



DIAMOND DESIGN 



lost, for we may take it that the spectator 
will not look at the rose from the side of 
the mounting (fig. 23). If, on the other 
hand, we flatten the apex of the rose (fig. 24) 
(which is the usual method), a leakage will 
occur through its base. There is, of course. 




no amelioration in the refraction if the 
light passes from one facet to another 
similarly placed (as shown in fig. 23, path 
S' P' Q' R' T') . Taking the effect as a whole, 
the least unsatisfactory shape is as shown 
in fig. 24, with the angles a about 49° and 
30° for the base and the apex respectively. 
The rose cut, however, is fundamentally 



MATHEMATICAL 



63 



wrong, as we have seen above, and should 
be abohshed altogether. It is the high 
cost of the material that is the cause of its 
still being used in cases where the rough 
shape is especially suitable, and then only 
in small sizes. In actual practice the 




Fig. 24. 

proportions of the cut rose depend largely 
upon those of the rough diamond, the 
stone being cut with as small a loss of 
material as possible. Generally the 
values of a are much below those given 
above, i.e. 49° and 30°, as where the 
material is thick enough to allow such 
steep angles it is much better to cut it 
into a brilhant. 



64 



DIAMOND DESIGN 



The Brilliant 

A . Back of the Brilliant 

Let us now pass to the consideration of 
the other alternative, i.e. where the top 




surface is a horizontal plane A B and 
where the bottom surface A C is inclined 
at an angle a to the horizontal (fig. 
25). As before, we have to introduce 
a third plane B C to have a symmetrical 
section. 



MATHEMATICAL 65 

' First Reflection 

Let a vertical ray P Q strike A B. As the 
angle of incidence is zero, it passes into the 
stone without refraction and meets plane 
A C at R. Let R N be the normal at that 
point, then, for total reflection to occur, 

angle NRQ== 24° 26'. 
But 

angle N R Q := angle Q A R === a, 

as AQ and QR, AR and RN are per- 
pendicular. 

Therefore, for total reflection of a vertical 
ray, 

a = 24° 26'. 

Let us now incline the ray P Q so that it 
gradually changes from a vertical to a 
horizontal direction, and let P' Q' be such 
a ray. Upon passing into the diamond it 
is refracted, and strikes AC at an angle 
Q' R' N' where R' W is the normal to A C. 
When P' Q' becomes horizontal, the angle 
of refraction T' Q' R' becomes equal to 
24° 26'. This is the extreme value attain- 

5 



66 DIAMOND DESIGN 

able by that angle ; also, for total reflection, 
angle Q' R' N' must not be less than 24° 26'. 
If we draw R' V, vertical angle V R' Q' = 
R' Q' T' = 24° 26', and 

angle V R' N' - V R' Q' + Q' R' N' 

-: 24° 26' + 24° 26' 
= 48° 52' 

as before, 

a = angle V R' N', 
and therefore 

a :^ 48° 52' . . . (9) 

For absolute total reflection to occur at the 
first facet, the inclined facets must make an 
angle of not less than 48° 52' with the horizontal. 

Second Reflection 

When the ray of light is reflected from 
the first inclined facet AC (fig. 26), it 
strikes the opposite one B C. Here too the 
light must be totally reflected, for other- 
wise there would be a leakage of light 
through the back of the gem-stone. Let us 
consider, in the first instance, a ray of light 
vertically incident upon the stone. The 



MATHEMATICAL 



67 



path of the ray will be P Q R S T. If R N 

and S N' are the normals at R and S respec- 
tively, then for total reflection, 

angle N' S R -= 24° 26'. 



P 



Q 



M 



B 



N 



o( 



N' 



\oC 



Fig. 26. 

Let us find the value of a to fulfil that 
condition : 

angle Q R N = angle Q A R = a 

as having perpendicular sides. 

angle S R N =:= angle Q R N 



68 DIAMOND DESIGN 

as angles of incidence and reflection. 
Therefore 

angle NRS = a. 
Now let 

angle N' S R = a; 

Then, in triangle R S C, 

angle S R C = 90° — a 
angle R S C = 90°— x 
angle RCS == 2 X angle RCM 

-=2 X ARQ =- 2(90°— a). 

The sum of these three angles equals 
two right angles, 

90°— a +90°— :\; +180°— 2 a = 180°, 
or 

3a -\-x = 180° 
3a = i8o°— X. 

Now, X is not less than 24° 26', therefore 
a is not greater than 

3 
Let us again incline P Q from the vertical 
until it becomes horizontal, but in this case 
in the other direction, to obtain the inferior 
limit. 



MATHEMATICAL 



69 



Then (fig. 27) the path will be PQRS. 
Let QT, RN, SN' be the normals at Q, R, 
and S respectively. At the extreme case, 
TQR will be 24° 26'. Draw RV vertical 
at R. 




Fig. 27. 

Then 

angle Q R V = angle T Q R = 24° 26' 
angle V R N = a. 
As before, in triangle R S C, 
angle SRC = go^-NRS = 9o°-a-24°26' 
angle RCS = 2(90°- a) 
angle R S C = 90°— x. 



70 DIAMOND DESIGN 

Then 

90— a— 24° 26'+i8o°— 2a+90°— ;t; = 180° 
3a +% = 180^-24° 26' - 155° 34' 

In the case now considered, 

X = 24° 26'. 
Then 

3a = 155° 34'-24' 26' = 131° 8' 
« == 43° 43' . • . . (10) 
For absolute total reflection at the second 
facet, the inclined facets must make an angle 
of not more than 43° 43' with the horizontal. 

We will note here that this condition 
and the one arrived at on page 66 are in 
opposition. We will discuss this later, and 
will pass now to considerations of refraction. 

Refraction 

First case : ql is less than 45° 

In the discussion of refraction in a 
diamond, we have to consider two cases, 
i.e. a is less than 45° or it is more than 45°. 
Let us take the former case first and let 
P Q R S T (fig. 28) be path of the ray. Then, 



MATHEMATICAL 



71 



if S N is the normal at S, we know that for 
total reflection at S angle RSN == 24° 26'. 
We want to avoid total reflection, for if 
the light is thrown back into the stone, 
some of it may be lost, and in any case the 




Fig. 28. 

ray will be broken too frequently and the 
result will be disagreeable. 
Therefore, 

angleRSN< 24°26' . (11) 

Suppose this condition is fulfilled and the 
light leaves the stone along S T. It is re- 
fracted, and its colours are dispersed into 



72 DIAMOND DESIGN 

a spectrum. It is desirable to have this 
spectrum as long as possible, so as to disperse 
the various colours far away from each other. 
As we know, this will give us the best 
possible ** fire." 

This result will be obtained when the 
ray is refracted through the maximum 
angle. Ry (ii) the value for that angle 
is 24° 26', and (11) becomes 

angle RSN == 24° 26' for maximum dis- 
persion. 

But then the light leaves A B tangentially, 
and the amount of light passing is zero. 
To increase that amount, the angle of 
refraction has to be reduced : the angle of 
dispersion decreases simultaneously, but 
the amount of light dispersed increases 
much more rapidly. Now we know that 
the angle of dispersion is proportional to 
the sine of the angle of refraction. It is, 
moreover, proved in optics that the amount 
of light passing through a surface as at 
AB is proportional to the cosine of the 
angle of refraction. The brilliancy pro- 



MATHEMATICAL ^z 

duced is proportional both to the amount 
of Hght and to the angle of dispersion, and 
therefore to their product, and (by the 
theory of maxima and minima) will be 
maximum when they are equal, i.e. when 
the sine and cosine of the angle of refraction 
are equal. For maximum brilliancy, there- 
fore, the angle of refraction should be 45°. 
This gives for angle R S N 

sin RSN = ^HI4_5 _ -7071 = .2930, 

2-417 2-417 

therefore 

angle RSN = 17° for optimum brilliancy (12) 

Let (fig. 28) Q X and R Y be the normals 
at Q and R respectively, and let ZZ' be 
vertical thr-ough R. 

We know that 

angle R Q X =- angle P Q X = a, 

therefore 

angle PQR == 2a. 

Produce Q R to Q'. 

Then, as P Q and Z Z' are parallel, 

angle Z R Q' = angle P Q R = 2a. 



74 DIAMOND DESIGN 

Now, let 

angle R S N = %( = 17° for optimum 
brilliancy) . 

Then, as Z Z' and S N are parallel, 

angle Z R S = a;. 

As they are complements to angles of 
incidence, 

angle Q R C = angle S R B = z (say), 

but 

angle Q' R B = angle Q R C, 

therefore 

angle S R Q' = 2i. 

In angle ZRQ' we have _.^" 

angle Z R Q' -- angle Z R S +angle S R Q' 

2a = X+2i , . (13) 

\ In triangle Q C R 

angle RCQ = 90°— a 

angle Q R C = ^ 

angle Q C R == 2(90°— a), 

therefore 

(90 — a) +z +180°— 2a = 180°, 



or 



i = 3a— 90°. 



MATHEMATICAL 75 

Introduce this value of i in (13), 

2a = A;+6a — 180° 
4a == 180°—% 

and giving x its value 17°, 

4a ^ i8o°— 17° = 163° 
a = 40° 45' . . . (14) 

If we adopt this value for a, the paths of 
oblique rays will be as shown in fig. 29, 
P Q R S T when incident from the left of the 
figure, and P' Q' R' S' T' when incident from 
the right.' Ray P Q R S T will leave the dia- 
mond after the second reflection, but with a 
smaller refraction than that of a vertically 
incident ray, and therefore with less **fire." 
Oblique rays incident from the left are, 
however, small in number owing to the acute 
angle Q R A with which they strike A C ; the 
loss of fire may therefore be neglected^____ 



Ray P'Q'R'S'r will strike AB at a 
greater angle of incidence than 24° 26', and 
will be reflected back into the stone. This 
is a fault that can be corrected by the 
introduction of inclined facets D E, F G ; 
ray P' Q' R' S' T' will then strike F G at an 



76 



DIAMOND DESIGN 



angle less than 24° 26', and this angle can 
be arranged by suitably inclining FG to 
the horizontal so as to give the best possible 
refraction. The amelioration obtained by 
thus taking full advantage of the refraction 

U 




is so great that the small loss of light caused 
by that arrangement of the facets is in- 
significant : the leakage occurs through the 
facet CB, near C, where the introduction 
of the facet D E allows hght to reach C B 
at an angle less than the critical. In a 



MATHEMATICAL 77 

brilliant, where CB is the section of the 
triangular side of an eight-sided pyramid, 
the area near the apex C is very small, and 
the leakage may therefore be considered 
negligible. 

Second case : a is greater than 45° 

In this case the path of a vertical ray 
will be as shown by P Q R S T in fig. 30, and 
the optimum value for a, which may be 
calculated as before, will be 

a=:49^i3' . . (15) 

As regards the vertical rays, this value 
gives a fire just as satisfactory as (14) 
(a = 40"^ 45') ; let us consider what happens 
to oblique rays. 

Rays incident from the left as pqrstu 
may strike B C at an angle of incidence less 
than the critical, and will then leak out 
backwards. Or they may be reflected along 
s t, and may then be reflected into the stone. 
Both alternatives are undesirable, but they 
do not greatly affect the brilliancy of the 
gem, because, as we have seen, the amount 
of light incident from the left is small. 



78 



DIAMOND DESIGN 



That incident from the right is, on the 
contrary, large. 

Let us follow ray P'Q'R'S'T'. It will 
be reflected twice, and will leave the 




diamond after the second reflection, like 
the vertically incident ray, but with a 
smaller refraction, and consequently less 
fire ; most of the light will be striking 
the face A B nearly vertically when leaving 
the stone, and the fire will be very small. 



MATHEMATICAL 79 

This time it is impossible to correct the 
defect by introducing accessory facets, as 
the paths S'T' of the various obHque rays 
are not locaHsed near the edge B, but are 
spread over the whole of the face ; we are 
therefore forced to abandon this design. 

Summary of the Results obtained for a 

We have found that — 
For first reflection, a must be greater 
than 48° 52'. 

For second reflection, a must be less than 

43° 43'- 

For refraction, a may be less or more 

than 45°. When more, the best value is 
49° 15', but it is unsatisfactory. When less, 
the best value is 40° 45', and is very satis- 
factory, as the light can be arranged to 
leave with the best possible dispersion. 

Upon consideration of the above results, 
we conclude that the correct value for a is 
40° 45', and gives the most vivid fire and 
the greatest brilliancy, and that although 
a greater angle would give better reflection, 
this would not compensate for the loss due 



So 



DIAMOND DESIGN 



B 



to the corresponding reduction in dispersion. 
In all future work upon the modern brilliant 
we will therefore take 

a = 40° 45'. 

B. Front of the Brilliant 

When arriving at the value of a = 40° 45', 
we have explained how the use of that 

angle introduced 
defects which 
could be cor- 
rected, by the use 
of extra facets. 
The section will 
therefore be 
shaped somewhat 
as in fig. 31. It will be convenient to give 
to the different facets the names by which 
they are known in the diamond-cutting 
industry. These are as follows : — 

A C and B C are called pavilions or quoins 
(according to their position relative 
to the axis of crystallisation of the 
diamond) . 



p . E 

c 



Fig. 31. 



MATHEMATICAL 8i 

AD and EB are similarly called bezels 

or quoins. 
DE is the table. 
FG is the culet, which is made very 

small and whose only purpose is to 

avoid a sharp point. 

Through A and B passes the girdle of 
the stone. 

We have to find the proportions and in- 
clination of the bezels and the table. These 
are best found graphically. We know that 
the introduction of the bezels is due to the 
oblique rays ; it is therefore necessary to 
study the distribution of these rays about 
the table, and to find what proportion of 
them is incident in any particular direction. 

Consider a surface A B (fig. 32) upon which 
a beam of light falls at an angle a. Let 
us rotate the beam so that the angle 
becomes ^ (for convenience, the figure 
shows the surface A B rotated instead to 
A' B, but the effect is the same). The light 
falling upon AB can be stopped in the 
first case by intercepting it with screen 



82 



DIAMOND DESIGN 




Fig. 32. 



B C, and in the second with a screen B C 
where B C C is at right angles to the 

direction of thebeam. 
And if the intensity 
of the light is uni- 
form, the length of 
B C and B C wiU be 
a measure of the 
amount of light fall- 
ing upon A B and A B' respectively. 
Now 

BC^ABsina 
BC'-3 A'BsiniS = ABsin^g. 

Therefore, other things being equal, the 
amount of light falling upon a surface is pro- 
portional to the sine of the angle between 
the surface and the direction of the light. 
We can put it as follows : — 

If uniformly distributed light is falling 
from various directions upon a surface A B, 
the amount of light striking it from any 
particular direction will be proportional 
to the sine of the angle between the surface 
and that direction. 



MATHEMATICAL 



83 



If we draw a curve between the amount 
of light striking a surface from any parti- 
cular direction, and the angle between the 
surface and that direction, the curve will 
be a sine curve (fig. 33) if the light is equally 
distributed and of equal intensity in all 
directions. 

M 




O 4-2 70''2 90 l09'/^ 158 180'' 

M' 

Fig. 33. 

For calculations we can assume this to 
be the case, and we will take the distri- 
bution of the quantity of light at different 
angles to follow a sine law. 

It is convenient to divide all the light 
entering a diamond into three groups, one 
of vertical rays and two of oblique rays, 
such that the amount of light entering 
from each group is the same. Now in the 



84 DIAMOND DESIGN 

sine curve (fig. 33) the horizontal distances 
are proportional to the angles between the 
table of a diamond and the direction of the 
entering rays ; the vertical distances are 
proportional to the amount of light entering 
at these angles. The total amount of light 
entering will be proportional to the area 
shaded. That area must therefore be 
divided into three equal parts ; this may 
be done by integrals, or by drawing the 
curve on squared paper, counting the 
squares, and drawing two vertical lines on 
the paper so that one-third of the number 
of the squares is on either side of each line. 
By integrals, 

area —/"sin xdx ^= — cos x. 
The total area = [— cos x] = i -fi =: 2, 



therefore 



"o area — "o. 



The value of a corresponding to the vertical 
dividing lines on the curve is thus given 

by 



cos x = I — I = I 
cos:^ = I— t-- — i 



MATHEMATICAL 85 

therefore 

X = 7o|° approximately 



and 



X = 1091°. 



Taking the value x = 90° as zero for 
reckoning the angles of incidence, 



and 



z == 90°— 701° -= 19J 
i = 90°— 1091° = — 19I, 



The corresponding angles of refraction are 

sin i sin iqi^ -333 3 
sin r = = ^^ = 0000 _ . J07 7 

;z 2-417 2-417 

r-7°52'. 
The range of the different classes is thus 
as follows : — 

Angle of incidence : 

vertical rays —19!° to +19!° 
oblique rays —90° to — I9j° 
and +191° to +90°. / 

Angle of refraction : 

vertical rays —7° 52' to +7° 52' 

oblique rays —24° 26' to —7° 52' 

and +f 52' to +24° 26'. 

The average angle of each of these classes 



86 DIAMOND DESIGN 

may be obtained by dividing each of the 
corresponding parts on the sine curve in two 
equal parts. The results are as follows : — 

Angle of incidence : 

vertical rays o° 
oblique rays —42° 
and +42°. 

Angle of refraction : 

vertical rays 0° 
oblique rays —16° 
and +16°. 

For the design of the table and bezels, 
we have to know the directions and positions 
of the rays leaving the stone. The values 
just obtained would enable us to do so if 
all the rays entering the front of the gem 
also left there. We have, however, adopted 
a value for a (a = 40° 45') which we know 
permits leakage, and we have to take that 
leakage into consideration. 

The angle where leakage begins is in- 
clined at 24° 26' to the pavilion (fig. 24). 
We have thus 

Q' R' W = 24° 26', 



MATHEMATICAL 87 

therefore 

Q' R' A' = 90°-24° 26'= 65° 34^ 
Now in triangle A Q' R', 

Q' R' A + A Q' R' +R' A Q' - 180°, 
therefore 

A Q' R' = 180° -65° 34-40° 45' 

= 73° 41'. 
The Hmiting angle of refraction R'Q'T 
is thus 

= 90°-73° 41' - 16° 19', 
corresponding to an angle of incidence of 

sini = nsinr = 2-417 sin 16° 19' 
= 2-417 X -281 = -678. 

i = 42^. 

Upon referring to the sine curve, we find 
that the area shaded (fig. 34), which repre- 
sents the amount of light lost by leakage, 
although not so large as if the same number 
of degrees leakage had occurred at the 
middle part of the curve, is still very ap- 
preciable, forming as it does about one- 
sixth of the total area. Just under one- 
half (exactly -493) of the fight incident 
obhquely from the right (fig. 25) is effective, 



88 



DIAMOND DESIGN 



the other half being lost by leakage. Still, 
the sacrifice is worth while, as it produces 
the best possible fire. 

The oblique rays incident from the right 
range therefore 191° to 42!°, with an average 
(obtained as before) of 30° 15'. The corre- 




/«o^ 



Fig. 34. 

sponding refracted rays are 7° 52', 16° 19', 
and 12° o'. 

We have now all the information necessary 
for the design of the table and the bezels. 

Design of Table and Bezels (fig. 35) 

Let us start with the fundamental section 
ABC symmetrical about MM', making the 
angles A C B and A B C 40° 45'. 

V 



[To face p. 88. 




E. & F. N. SPON, LTD., LONDON. 



[To face p. 88. 



A- 



P P, P' P, P 



2 »^3 




M 



R 



Fig. 35. 



Tolkowsky, Diamond Design.] 



E. & F. N, SPON, LTD., LONDON. 



MATHEMATICAL 89 

The bezels have been introduced into the 
design to disperse the rays which were 
originally incident from the right upon the 
facet AB. To find the limits of the table, 
we have therefore to consider the path of 
limiting oblique ray. We know that this 
ray has an angle of incidence of 42^° and 
an angle of refraction of 16° 19'. Let us 
draw such a ray P Q : it will be totally 
reflected along Q R, if we make P Q N = 
N Q R, where Q N is the normal. Now Q R 
should meet a bezel. 

If the ray PQR was drawn such that 
M P = M R, then P and R will be the points 
at which the bezels should meet the table. 
For if PQ be drawn nearer to the centre of 
the stone, QR will then meet the bezel, 
and if P Q be drawn further away, it will 
meet the opposite bezel upon its entry into 
the stone and will be deflected. 

The first point to strike us is that no 
oblique rays incident from the left upon 
the table strike the pavilion AB, owing to 
the fact that the table stops at P. We 
will, therefore, treat them as non-existent, 



90 DIAMOND DESIGN 

and confine our attention to the vertical 
rays and those incident from the right. 

Let us draw the limiting average rays 
of these two groups, i.e. the rays of the 
average refractions o° and 12° passing 
through P, P S, and P T. The length of the 
pavilion upon which the rays of these two 
groups fall are thus respectively C S and C T. 

The rays of the first group P' Q' R' S' are 
all reflected twice before passing out of 
the stone, and make, after the second 
reflection, an angle of 17° with the vertical 
(as by eq. (12)). Of the rays of the second 
group, most are reflected once only (Pj Qi Rj) 
and make then an angle of 69^° with the 
vertical (this angle may be found by 
measurement or by calculation). Part 
of the second group is reflected twice 
(P3Q3R3S3), and strikes the bezel at 29° 
to the vertical. This last part will be 
considered later, and may be neglected 
for the moment. 

We have to determine the relation be- 
tween the amount of light of the first 
group and of the first part of the second 



MATHEMATICAL 91 

group. Now we know that the amount 
of obHque Ught reflected from a surface on 
paviHon A C is -493 of the amount of verti- 
cal light reflected (cp. fig. 34 and context). 
If we take as limit for the once-reflected 
oblique ray the point E (as a trial) on 
pavilion B C, i.e, if it is at E that the girdle 
is situated, then the corresponding point 
of reflection for that oblique ray will be 
Q2 (fig. 35). The surface of pavilion upon 
which the oblique rays then act will be 
limited by S and Q2, and as in a brilliant 
the face A C is triangular, the surface will 
be proportional to 

sc2-q;c^ 

'Similarly, the surface upon which the 
vertical group falls will be proportional to 



TO. 

Thus we have as relative amounts of 
Hght— 

for vertical rays TC^ 

for oblique rays -493(80^— QO). 

The first group strikes the bezel at 17° 



92 DIAMOND DESIGN 

to the vertical, and the second at 69!° to 
the vertical. The average inclination to 
the vertical will thus be 



17 xTg+691 x^93(Sg--QC^) 
TC^+.493(SC2-QC^) 

Let us draw a line in that direction 
(through R, say), and let us draw a perpen- 
dicular to it through R, R E ; then that 
perpendicular will be the best direction for 
the bezel, as a facet in that direction takes 
the best possible advantage of both groups 
of rays. 

If the point E originally selected was not 
correct, then the perpendicular through R 
will not pass through E, and the position 
of E has to be corrected and the corre- 
sponding value of CQ2 correspondingly 
altered until the correct position of E is 
obtained. 

For that position of E (shown on fig. 35), 
measures scaled off the drawing give 

CS =- 2-67 CS2 = 7-12 __ 

CT =2-13 GT^ = 4-54 CS^-CQ^ = 4-57. 
CQ, = i-6o CQ,^ = 2-56 



MATHEMATICAL 93 

Therefore the average resultant incHna- 
tion will be 

17 X CT^+691 X^3(Cg-CQ^) 
(CT2+.493(CS^-CQ^) 
_ 17 X 4-54 +69-5 X -493 X 4-57 
4-54 +-493x4-57 

to the vertical. 

By the construction, the angle /S, i.e. the 
angle between the bezel and the horizontal, 
has the same value 

^ = 34r-. 

The small proportion of oblique rays 
which are reflected twice meet the bezel 
near its edge, striking it nearly normally : 
they make an angle of 29° with the vertical. 
Facets more 'steeply inclined to the hori- 
zontal than the bezel should therefore be 
provided there. The best angle for re- 
fraction would be 29° +17° == 46°, but if 
such an angle were adopted most of the 
light would leave in a backward direction, 
which is not desirable. It is therefore 



94 DIAMOND DESIGN 

advisable to adopt a somewhat smaller 
value ; an angle of about 42° is best. 

Faceting / 

The faceting which is added to the 
brilliant is shown in fig. 43. Near the 
table, '* star " facets are introduced, and 
near the girdle, ''cross'' or *' half '' facets 
are used both at the front and at the back 
of the stone. 

We have seen that it is desirable to intro- 
duce near the girdle facets somewhat steeper 
than the bezel, at an angle of about 42°, 
by which facets the twice-reflected oblique 
rays might be suitably refracted. The 
front " half " facets fulfil this purpose. 

We have remarked that the angle (42°) 
had to be made smaller than the best angle 
for refraction (46°) to avoid light being 
sent in a backward direction, where it is 
unlikely to meet either a spectator or a 
source of light. 

To obviate this disadvantage, a facet two 
degrees steeper than the pavilion should 
be introduced near the girdle on the back 



MATHEMATICAL 95 

side of the stone ; for then the second 
reflection of the obHque rays will send them 
at an angle of 25° to the vertical (instead 
of 29°), and the best value for refraction 
for the front half facets will be between 

25°+i7°-42°. 

These values are satisfactory also as 
regards the distribution of light ; for now 
the greater part of the light is sent not in 
a backward, but in a forward, direction. 

The facet two or three degrees steeper 
than the pavilion is obtained in the brilliant 
by the introduction of the back '* half '' 
facet, which is, as a matter of fact, generally 
found to be about 2° steeper than the 
pavilion in well-cut stones. Where the 
cut is somewhat less fine and the girdle is 
left somewhat thick (to save weight), that 
facet is sometimes made 3° steeper, or 
even more, than the pavilion. 

The ''star'' facet was probably intro- 
duced to complete the design of the brilliant, 
which without its use would be lacking in 
harmony, but which its introduction makes 



96 DIAMOND DESIGN 

exceedingly pleasing from the point of view 
of the balance of lines. 

Let us examine the optical consequences 
of the use of '' star " facets. 

On the one hand, their inclination — 
about 15° to the horizontal — permits a 
certain amount of light to leave the stone 
without being sufficiently refracted. On 
the other, they diminish the area of the 
bezels and consequently decrease the leak- 
age of light which occurs through the bezel 
and the opposite pavilion (owing to the 
surfaces being nearly parallel). They also 
cause a somewhat better distribution of 
light, for they deflect part of the rays 
which would otherwise have increased the 
strength of the spectra refracted by the 
bezels, and create therewith spectra along 
other directions ; it is true that, as seen 
above, these spectra will be shorter. But 
they will be more numerous ; and though 
the " fire " — as consequent from the great 
dispersion of the rays of light — will be 
slightly diminished, the "life" — if we may 
term '' life '' the frequency with which a 



MATHEMATICAL 97 

single source of light will be reflected and 
refracted to a single spectator upon a 
rotation of the stone — ^will be increased to 
a greater degree. And if we take into ac- 
count the decrease in the leakage of light, 
we may conclude that the, introduction of 
the stars, on the whole, is decidedly ad- 
vantageous in the brilliant. 

Best Proportions of a Brilliant 

We have thus as best section of a brilliant 
one as given in fig. 35, A B C D E, where 

a = 40° 45' 

^ - 34' 30'. 

D E is obtained from P R in fig. 35. 

If we make the diameter A B of the stone 
100 units, then the main dimensions are 
in the following proportions (fig. 35) : — 

Diameter A B . . .100 
Table D E . . . . 53-0 
Total thickness M C . . 59-3 
Thickness above girdle MM' 16-2 
„ below „ M'C . 43-1 

• 7 



98 



DIAMOND DESIGN 



Fig. 36 shows the outline of a brilliant 
with these proportions. 

These proportions can be approximated 
as follows : — 

In a well-cut brilliant the diameter of 







/OO'O 








S3'0 














J 

5$ 


i 


y^ 


\ 






/ 













Fig. 36. 

the table is one-half of the total diameter, 
and the thickness is six-tenths of the total 
diameter, rather more than one-quarter 
of the thickness being above the girdle and 
rather less than three-quarters below. 

It is to be noted here that a different pro- 
portion is generally stated for the thickness 



MATHEMATICAL 



99 





Fig. 37. 



100 DIAMOND DESIGN 

above the girdle (*' one- third of the total 
thickness''), both in works upon diamonds 
and by diamond poHshers themselves. It 
is true that diamonds were cut thicker 
above the girdle and with a smaller table 
before the introduction of sawing, for 
then the table was obtained by grinding 
away a corner or an edge of the stone, and 
the loss in weight was thus considerable, 
and would have been very much greater 
still if the calculated proportions had been 
adopted. With the use of the saw, the loss 
in weight was enormously reduced and the 
manufacture of sawn stones became there- 
fore much finer and more in accordance 
with the results given above. It is a 
remarkable illustration of conservatism that 
although diamonds have been cut for de- 
cades with J (approximately) of the thick- 
ness above the girdle, yet even now the 
rule is generally stated as | of the 
thickness. 

Stones are still cut according to that 
rule, but then they are not sawn stones as 
a rule, and the thickness is left greater 



MATHEMATICAL loi 

to diminish the loss in weight. The 
brilliancy is not greatly diminished by 
making the stone slightly thicker over the 
girdle. 

Comparison of the theoretically best 
Valuer with those used in Practice 

In the course of his connection with the 
diamond-cutting industry the author has 
controlled and assisted in the control of 
the manufacture of some million pounds' 
worth of diamonds, which were all cut 
regardless of loss of weight, the only aim 
being to obtain the liveliest fire and the 
greatest brilliancy. The most brilliant 
larger stones were measured and their 
measures noted. It is interesting to note 
how remarkably close these measures, which 
are based upon empirical amelioration and 
rule-of -thumb correction, come to the 
calculated values. 

As an instance the following measures, 
chosen at random, are given (the dimensions 
are in millimetres) : — 



102 



DIAMOND DESIGN 



Table I 



a . 


4or 


4or 


40° 


41° 


41° 


^ • . 


35° 


35° 


34r 


33° 


34° 


AB . 


7-00 


7-o8 


6-50 


21-07 


9-12 


MC . 


4'12 


4-35 


3-6i 


12-34 


5-47 


MM' . 


i-o8 


1-32 


0-85 


3-31 


i-6i 



These measures, worked out in percentage 
of A B, give : — 



Table II 



a 


40f° 


4or 


40° 


41° 


41° 


40^42' 


40^45' 


i8 . 


35° 


35° 


344° 


33° 


34° 


34° 18' 


34° 30' 


AB 


100 


100 


100 


100 


100 


100 


100 


MC 


587 


61-4 


55-4 


58-5 


60 


58-9 


59-3 


MM' 


157 


i8-6 


13-3 


157 


17-8 


l6-2 


l6-2 


M'C 


43-0 


42-8 


42-1 


42-8 


42-2 


42-6 


43-1 



In the seventh column the averages of 
the measures are worked out, and the eighth 
gives the calculated theoretical values. It 
will be noted that the values of a, j8, and 
M M' correspond very closely indeed, but 



MATHEMATICAL 103 

that M C and M' C are very slightly less than 
they should be theoretically. 

The very sHght difference between the 
theoretical and the measured values is due 
to the introduction of a tiny facet, the 
collet, at the apex of the pavihons. This 
facet is introduced to avoid a sharp point 
which might cause a split or a breakage 
of the diamond. 

What makes the agreement of these 
results even more remarkable is that in the 
manufacture of the diamond the pohshers 
do not measure the angles, etc., by any 
instrument, but judge of their values en- 
tirely by the eye. And such is the skill 
they develop, that if the angles of two 
pavihons of a briUiant be measured, the 
difference between them will be in- 
appreciable. 

We may thus say that in the present-day 
well-cut brilHant, perfection is practically 
reached : the high-class brilliant is cut as 
near the theoretic values as is possible in 
practice, and gives a magnificent brilliancy 
to the diamond. 



104 DIAMOND DESIGN 

That some new shape will be evolved 
which will cause even greater fire and life 
than the brilliant is, of course, always 
possible, but it appears very doubtful, and 
it seems likely that the brilliant will be 
supreme for, at any rate, a long time yet. 



(PR. 1605.) 



PRINTED IN GREAT BRITAIN BY NEILL AND CO., LTD., EDINBURGH. 



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